5. Solve the simultaneous equations
$$\begin{gathered}
x - 2 y = 1 , \\
x ^ { 2 } + y ^ { 2 } = 29 .
\end{gathered}$$
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Question 5:
Answer Marks
Guidance
Answer Mark
Guidance
\(x = 1 + 2y\) and substitute \(\to (1+2y)^2 + y^2 = 29\) M1
Attempt to sub leading to equation in 1 variable
\(\Rightarrow 5y^2 + 4y - 28 (= 0)\) A1
Correct 3TQ (condone \(= 0\) missing)
\((5y + 14)(y - 2) = 0\) M1
Attempt to solve 3TQ leading to 2 values for \(y\)
\(y = 2\) or \(-\frac{14}{5}\) (both) A1
Condone mislabelling \(x =\) for \(y = \ldots\)
\(y = 2 \Rightarrow x = 5\); \(y = -\frac{14}{5} \Rightarrow x = -\frac{23}{5}\) M1A1 f.t. (6)
Attempt at least one \(x\) value; f.t. in \(x = 1 + 2y\); both values
Note: False squaring (e.g. \(y = x^2 + 4y^2 = 1\)) can only score last 2 marks.
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## Question 5:
| Answer | Mark | Guidance |
|--------|------|----------|
| $x = 1 + 2y$ and substitute $\to (1+2y)^2 + y^2 = 29$ | M1 | Attempt to sub leading to equation in 1 variable |
| $\Rightarrow 5y^2 + 4y - 28 (= 0)$ | A1 | Correct 3TQ (condone $= 0$ missing) |
| $(5y + 14)(y - 2) = 0$ | M1 | Attempt to solve 3TQ leading to 2 values for $y$ |
| $y = 2$ or $-\frac{14}{5}$ (both) | A1 | Condone mislabelling $x =$ for $y = \ldots$ |
| $y = 2 \Rightarrow x = 5$; $y = -\frac{14}{5} \Rightarrow x = -\frac{23}{5}$ | M1A1 f.t. (6) | Attempt at least one $x$ value; f.t. in $x = 1 + 2y$; both values |
**Note:** False squaring (e.g. $y = x^2 + 4y^2 = 1$) can only score last 2 marks.
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5. Solve the simultaneous equations
$$\begin{gathered}
x - 2 y = 1 , \\
x ^ { 2 } + y ^ { 2 } = 29 .
\end{gathered}$$
\hfill \mbox{\textit{Edexcel C1 2005 Q5 [6]}}